The nilpotent subvariety of the vector space associated to a symmetric pair (Q761513)

Let \({\mathfrak G}\) be a complex simple Lie algebra, G the adjoint group and t a linear involution of \({\mathfrak G}\). We note \({\mathfrak K}:=\{X\in {\mathfrak G}| t(X)=X\},\) K the corresponding subgroup of G and \(V: =\{X\in {\mathfrak G}| t(X)=-X\}.\) The author studies the (affine algebraic) variety \({\mathcal N}(V)\) of nilpotent elements of \({\mathfrak G}\) contained in V and the action of \({\mathfrak K}\) on \({\mathcal N}(V)\). For example, he determines the number of irreducible components of \({\mathcal N}(V)\) and constructs a resolution of \({\mathcal N}(V)\). The case ''\({\mathfrak G}\) of type \(A_ n''\) is closely examined.